From [1]}, [2]}, [3]},
we have that the eigenfunctions \(\psi _k\)  and eigenvalues \(\lambda _k\)  of the operator \(\mathcal {L}\)  (REF ) are given by
\(\begin{aligned}\psi _k(s) &= \frac{\widehat{\psi }_k(s)}{\Vert \widehat{\psi }_k\Vert _{L^2}}\,, \qquad \lambda _k = \alpha _k^4\,, \qquad k=1,2,\dots \\&\text{where } \cos (\alpha _k)\cosh (\alpha _k)=1\,, \quad \alpha _0=0 \\&\text{and } \widehat{\psi }_k(s) = \left(\cos (\alpha _k)-\cosh (\alpha _k)\right) \left(\cos (\alpha _k s)- \cosh (\alpha _k s) \right) \\&\hspace{56.9055pt} + \left(\sin (\alpha _k) +\sinh (\alpha _k)\right) \left(\sin (\alpha _k s)-\sinh (\alpha _k s)\right)\, .\end{aligned}\) 
